The use of robust formulations to guard against overfitting and to guarantee out-of-sample performance was popularized by [1]}. During the last decade in particular, Distributionally Robust Optimization (DRO) formulations have witnessed a surge in popularity.
Such formulations consider an ambiguity set \(\mathcal {U}_T(\hat{\mathbb {P}}_T)\subseteq \mathcal {P}\)  around the empirical distribution \(\hat{\mathbb {P}}_T\)  and protect against overfitting by considering decisions which minimize the worst-case cost over all considered probability distributions in \(\mathcal {U}_T(\hat{\mathbb {P}}_T)\)  instead of merely the empirical one.
That is, they estimate the cost of any decision as
\(\sup _{\mathbb {P}^{\prime } \in \mathcal {P}} \left\lbrace  \mathbb {E}_{\mathbb {P}^{\prime }}(\ell (x,\xi ))\ : \ \mathbb {P}^{\prime } \in \mathcal {U}_T(\hat{\mathbb {P}}_T) \right\rbrace .\) 
